# Ordinary deformations are unobstructed in the cyclotomic limit

## TL;DR

This paper proves that under certain conditions, the deformation ring of ordinary Galois representations in the cyclotomic limit is free over the Witt vectors, extending understanding of deformation theory in number theory.

## Contribution

It establishes that the deformation ring in the cyclotomic limit is free over Witt vectors, assuming Noetherianity and vanishing of specific invariants, which was previously conjectural.

## Key findings

- Deformation ring $R_
$ is free over Witt vectors under certain conditions.
- Noetherianity and vanishing of $$-invariants are key assumptions.
- Results extend the understanding of Galois deformation theory in the cyclotomic setting.

## Abstract

The deformation theory of ordinary representations of the absolute Galois groups of totally real number fields (over a finite field $k$) has been studied for a long time, starting with the work of Hida, Mazur and Tilouine, and continued by Wiles and others. Hida has studied the behaviour of these deformations when one considers the $p$-cyclotomic tower of extensions of the field. In the limit, one obtains a deformation ring $R_\infty$ classifying the ordinary deformations of the (Galois group of) the $p$-cyclotomic extension. We show that if $R_\infty$ is Noetherian and certain adjoint $\mu$-invariants vanish (as is often expected), then $R_\infty$ is free over the ring of Witt vectors of $k$.